Bound states for rapidly oscillatory Schrödinger operators in dimension 2

Abstract

We study the eigenvalues of Schrödinger operators on ${\mathbb{R}}^2$ with rapidly oscillatory potential $V(x)=W(x,x/\epsilon )$, where $W(x,y)\in C_0^{\infty }({\mathbb{R}}^2\times {\mathbb{T}}^2)$ satisfies ${\int }_{{\mathbb{T}}^2}W(x,y)dy=0$. We show that for $\epsilon$ small enough, such operators have a unique negative eigenvalue, that is exponentially close to $0$.

11 pages; the second version contains simplifications